Regularity of area minimizing currents. II: Center manifold.

*(English)*Zbl 1345.49052This is the second part of a series of three papers on the regularity of area minimizing integer rectifiable currents. Here the authors obtain one of the main results in the analysis of the singularities, namely, the construction of the center manifold. In the case of higher codimension, singularities of currents can appear as higher order perturbation of smooth minimal submanifolds. Following the pioneering work of F. J. Almgren jun. [Almgren’s big regularity paper. \(Q\)-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. Edited by V. Scheffer and Jean E. Taylor. Singapore: World Scientific (2000; Zbl 0985.49001)], the authors approximate the minimizing current with the graph of a multiple valued function on the normal bundle of a suitable curved manifold. Such a manifold must be close to the “average of sheets” of the current. This property could guarantee a singular first order expansion of the corresponding approximating map. In the present paper, the authors provide a construction of a center manifold \(\mathcal{M}\) and of an associated approximation of the corresponding area minimizing current via a multiple valued function \(F:\mathcal{M}\to \mathcal{A}_Q(\mathbb {R}^{m+n})\). The respective multiple valued function is defined at an appropriate scale which varies locally. Around any given point such a scale should be the first at which the sheets of the current cease to close. This leads to a Whitney-type decomposition of the reference \(m\)-plane where the refining algorithm is stopped. In each cube of the decomposition, the center manifold should be a smoothing of the average of the Lipschitz multiple valued approximation constructed in the first part of this article [the authors, Geom. Funct. Anal. 24, No. 6, 1831–1884 (2014; Zbl 1307.49043)] performed in a suitable orthonormal system of coordinates which changes from cube to cube. The authors obtain for the center manifold some \(C^{3,k}\)-estimates, which are the natural outcome of some Schauder estimates. If the current has multiplicity one everywhere, then the center manifold coincides with it, and hence one directly obtains a higher regularity than the one given by the usual De Giorgi-type (or Allard-type) argument.

For a given center manifold, it is possible to obtain a multivalued map on its normal bundle which approximates the current. The relevant estimates on this map and its approximation properties are then given locally for each cube of the Whitney decomposition used in the construction of the center manifold. At each scale where the refinement of the Whitney decomposition has stopped, the image of the obtained function coincides with the Lipschitz multiple valued approximation constructed in the first part of the present article. It is the same map whose smoothed average has been used to construct the center manifold. As a result, the graph of \(F\) is well centered, i.e. the average of \(F\) is very close to being the manifold \(\mathcal{M}\) itself. The regularity of the center manifold \(\mathcal{M}\) and the centering of the approximation map \(F\) are not the only properties needed to obtain the final proof (that will be given in the third part of this paper). Assume that around a certain point, at all scales larger than a given one, say \(s\), the excess decays and the sheets stay very close. If at scale \(s\) the excess is not decaying anymore, then the sheets must separate as well. In other words, if the tilting of the current is under control up to scale \(s\), the current must be in some sense “split before tilting”.

Now, we present a short description of the structure of the paper. First, the authors describe the construction algorithm and state the main existence theorem for the center manifold \(\mathcal{M}\). Then they obtain the \(\mathcal{M}\)-normal approximation and related estimates. Next, they present some additional conclusions upon \(\mathcal{M}\) and the \(\mathcal{M}\)-normal approximation. The construction of the center manifold is presented in some detail, by obtaining three key construction estimates. Then the existence result and the estimates for the \(\mathcal{M}\)-normal approximation are established. Then the authors obtain the separation and splitting before tilting. At the end, they study the persistence of \(Q\)-points and the comparison between different center manifolds. In appendix A they talk about the height bound revisited, in appendix B about changing coordinates for classical functions, in appendix C about two interpolation inequalities and in appendix D about the proof of Lemma 5.6 used in the paper. Editorial remark: for parts I and III, see [the authors, Geom. Funct. Anal. 24, No. 6, 1831–1884 (2014; Zbl 1307.49043); Ann. Math. (2) 183, No. 2, 499–575 (2016; Zbl 1345.49053)].

For a given center manifold, it is possible to obtain a multivalued map on its normal bundle which approximates the current. The relevant estimates on this map and its approximation properties are then given locally for each cube of the Whitney decomposition used in the construction of the center manifold. At each scale where the refinement of the Whitney decomposition has stopped, the image of the obtained function coincides with the Lipschitz multiple valued approximation constructed in the first part of the present article. It is the same map whose smoothed average has been used to construct the center manifold. As a result, the graph of \(F\) is well centered, i.e. the average of \(F\) is very close to being the manifold \(\mathcal{M}\) itself. The regularity of the center manifold \(\mathcal{M}\) and the centering of the approximation map \(F\) are not the only properties needed to obtain the final proof (that will be given in the third part of this paper). Assume that around a certain point, at all scales larger than a given one, say \(s\), the excess decays and the sheets stay very close. If at scale \(s\) the excess is not decaying anymore, then the sheets must separate as well. In other words, if the tilting of the current is under control up to scale \(s\), the current must be in some sense “split before tilting”.

Now, we present a short description of the structure of the paper. First, the authors describe the construction algorithm and state the main existence theorem for the center manifold \(\mathcal{M}\). Then they obtain the \(\mathcal{M}\)-normal approximation and related estimates. Next, they present some additional conclusions upon \(\mathcal{M}\) and the \(\mathcal{M}\)-normal approximation. The construction of the center manifold is presented in some detail, by obtaining three key construction estimates. Then the existence result and the estimates for the \(\mathcal{M}\)-normal approximation are established. Then the authors obtain the separation and splitting before tilting. At the end, they study the persistence of \(Q\)-points and the comparison between different center manifolds. In appendix A they talk about the height bound revisited, in appendix B about changing coordinates for classical functions, in appendix C about two interpolation inequalities and in appendix D about the proof of Lemma 5.6 used in the paper. Editorial remark: for parts I and III, see [the authors, Geom. Funct. Anal. 24, No. 6, 1831–1884 (2014; Zbl 1307.49043); Ann. Math. (2) 183, No. 2, 499–575 (2016; Zbl 1345.49053)].

Reviewer: Vasile Oproiu (Iaşi)

##### MSC:

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

49Q20 | Variational problems in a geometric measure-theoretic setting |

49N60 | Regularity of solutions in optimal control |

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\textit{C. De Lellis} and \textit{E. Spadaro}, Ann. Math. (2) 183, No. 2, 499--575 (2016; Zbl 1345.49052)

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